There are two distinct perspectives on the mathematical treatment of Riemannian manifolds. The intrinsic point of view uses only the local structure inherent to Riemannian manifolds to perform calculations. Alternatively, an extrinsic perspective isometrically depends upon an isometric embedding of an abstract Riemannian manifold into a Euclidean space as a means to concretely understand it. As proven by Nash, Riemannian manifolds can always be isometrically mapped into some Euclidean space, and therefore such an embedding must always exist [1]}. However, this embedding is not unique, and different embeddings can lead to different results for calculations on manifolds. As an example, the intrinsic distance between two points on a manifold, called the Riemannian distance, would depend upon the Riemannian metric determined when defining the manifold, while the extrinsic distance depends on the distance metric defined in the ambient Euclidean space and the specific embedding used [2]}.
